On polarizing filters application to apodization problems

Optica Applicata, Vol. X I I I , No. 4, 1983

On polarizing filters application

to apodization problems

Wa c ł a w Ur b a ń c z y k

Institute of Physics, Technical University of Wrocław, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland.

In the present work the possibility of employing the polarizing filters as apodizers in perfect imaging systems is studied. Using a numerical optimization technique, the discrete polarizing filters assuring the required Rayleigh or Sparrow resolution and the maximum encircled energy were found. It has been shown additionally that the polarizing filters possess some dynamic properties, i.e., that it is possible to change the Rayleigh or Sparrow resolution of the system solely by changing the position o f the filter. The polarizing filter performing (depending on its position) both demanded Rayleigh resolution and maximum encircled energy simultaneously was also determined.

1 . Introduction

The solutions of the most typical apodization problems b y em ploying the phase-amplitude filters may be found in papers [1-5]. Using the phase-amplitude filters as apodizers one may face the following difficulties :

1. There exist some apodization criteria which cannot be satisfied b y one filter, at the same time. F or example, the simultaneous radical improvement o f the resolution power o f the system and the encircled energy cannot be achie­ ved fo r phase-amplitude filter.

2. Phase-amplitude filters possess no dynamic properties. F or example, one filter may assure only one value of the system resolution power which cannot be increased unless the filter is changed.

These difficulties can be partly overcom e b y employing polarizing filters as apodizers.

2 . Polarizing filter

Let us assume that a polarizing filter composed of two elements F and G is inserted in the exit pupil of the perfect incoherent imaging system (Fig. 1).

* This paper has been presented at the European Optical Conference (EOC'83), May 30-June 4, 1983, in Rydzyna, Poland.

498 W . Ur b ań czyk

The following notation will be used: a — radius o f the exit pupil of the system, N — number of zones of the F element, B j = (la)/N — radius of the zone number I , ag — transmission angle of the G element, a j — transmission angle of the zone number I , /5 — angle describing the F element rotation.

the exit pupil of the system

Assuming additionally that the light passing through the system is polarized linearly, perpendicularly to the % -direction, we obtain the following formulae for the intensity spread function and the Strehl number of the system

N

I S F ( a , fi,p) = 0.25J0j j r ' B K j ( p ) [cos (2 ^ ) cos (2/9) —sin(2a2)sin (2/3)] |2,

r=i ’ (la) N

BN'(a,

/9) = 0.25 # # 2(0) [cos(2a2)cos(2/9)-sin(2a2)sin(2/9)]}2 (lb) I = N ’ where m e a p) = B l - A A z i H Bj _,J (lc )

On polarizing filters application... 499

I 0 — intensity at the central point of the diffraction spot for the system without the polarizing filter).

The effect of the polarizing filter on the I S F of the system is two fold. First, the intensity diminishes uniformly at each point of the image plane, secondly, the course of the I S F is changed. Since the first effect can be fully compensated b y increasing the intensity of light passing through the system, it is advisable to introduce the normalized Strehl number of the system

N

SN{a,P) = ^ 'it!F 'j(0 )[cos(2 a J)cos(2/S)-sin(2aJ)sin(2/S)]}2. (2)

/ = 1

3 . Rayleigh and Sparrow resolutions o f the system

A question of the Rayleigh resolution improvement may be formulated as the following optimization problem

S N ( d , ^ ) = m a x , (3a)

under the limiting condition

I S F (a, fi, 8r ) = 0 , fo r dR < 3.863. (3b)

Using a numerical optimization technique the values of parameters ax ... aN satisfying the conditions (3a) and (3b) have been found. W ithout loss of genera­ lity it has been assumed that parameter (9 equals zero.

Let us note that for any values of parameters ax ... aN and dR the condition (3b) may be fulfilled, only b y choosing parameter /5 according to the equation

70°·

SO0·

30°-10° Cf a ? '■ a<- fa

3.8

3A

3.0

2.6

2.2

-F ig. 2. The ax, a2, a3, a4 angles of the optimal polarizing filter vs. Rayleigh {SR) resolution o f the system (a), and Sparrow (Ss ) resolution of the system (b)

W . Uk ba n c zy i

Fig. 3. Strehl number 8N of the system for optimal polarizing filter assuring a given Rayleigt (dR) resolution (a), and Sparrow (Ss ) resolution (b). Strehl numbers vs. Rayleigh (a) and Sparrow (b) resolutions obtainable by rotation of the filter, optimal for ds = 3.15, 2.5£ (a) and Sg — 2.6, 2.3 (b) are also presented

Of course, condition (3a) is no longer satisfied. It means that the polariz­ ing filter, which is optimal (SN = max, /3 = 0) for a given Eayleigh resolution, may be also used to assure another (lower or higher) value of the system

reso-On polarizing filters'application... 501

lution power. Only a proper choice (Eq. (4)) of the /S angle, determining the position of the F element of the filter, is required. The computations have been performed for N = 4. Results, concerning both the Rayleigh and the Sparrow resolutions, are shown in Bigs. 2 -4 .

4 . Maximum encircled energy criterion

The encircled energy factor K ( â , ft) is usually defined as the ratio of the energy inside a circle o f radius q around the centre of the diffraction pattern (Fe) to the

total energy in the pattern (ET) K e(â, fi) where E e ( *, P) F T( a , p ) ’ (5) e JBe {â,i3) = 2 Jt f I 8 F ( p ) p d p , 0 F T(d, (3) = 2n j I 8 F ( p ) p d p , 0

Assuming that ft — 0 we obtain

N F e(d) = £ CIjCOS(2aI)cos{2aJ), i,j= i where C%I J 4 n J B K j ( p ) B K j { p ) p d p , for 1 # J o Q 27i J B K j ( p ) p d p , fo r I = J , 0 and (6>

(

7

)

N F T{a) = ^ P 1cos{2aI ), (8) J=1 where P j = T t i B ^ B W i

-Maximization of the encircled energy factor K e(a) m ay be represented as the following optimization problem

502 W . Urban czyk

under the limiting condition

K E a{d) = m ax. (9b)

^Numerical solution has been found for N = 4. Functions 8N(a) and K E e{a) are insensitive to the signs of parameters a1} a 2, as , a4. Then, there exist eight equivalent solutions satisfying the condition (9). Eesults are shown in Figs. 5-7.

100 80-KEp[%] ' A // / 60 _// a ■ ■ /// 50° 40- !t lorj / / / 30°· ^ " ior3l___ -20 r tar2l . . . 9 10°-" ' . ^ ' 0O l 0.2. 0.4 0.6 05 10 02 0.4 06 OS 1.0

Fig. 5. Tlie encircled energy factor K E e for diffraction limited systems : (A — nonapodized, B — apodized) as a function of radius q (g = 1 deno tes the first dark ring o f the nonapodized diffraction pattern)

Fig. 6. Dependence of the a,, a2, a3, a4 angles upon the radius q for the polarizing filter fulfilling the maximum encircled energy crite­ rion 0.8 -

0.4-0

.

2

-02 04 06 08 Ï0

£

Fig. 7. Dependence of the Strehl numbers SN upon the radius q for the polarizing filter fulfil­ ling the maximum encircled energy criterion

On polarizing filters application... 503 5 . Encircled energy factor and Rayleigh resolution o f the system

A given Eayleigh resolution of the system m ay be achieved for any values of parameters a1 . . . aN only b y a proper choice of the rotation angle /3 (Eq. (4)). Then, the filter fulfilling the maximum encircled energy criterion (for /3 = Q) will improve, after the F element rotation, the Eayleigh resolution o f the system. The rotation angles (/3) presented in Fig. 8, are needed to assure a given Eayleigh resolution realized b y different filters satisfying the m a x im um encircled energy criterion (e = 1 ) . Figure 9 shows the Strehl numbers of the system after the /?

Fig. 8. Rotation angles /3 needed to assure a given Rayleigh resol­ ution by the filters satisfying the maximum encircled energy criterion for q = 1 (ax = 0, a2 = ±16.2, as =

= ±27.01, a4 = ±35.82). Signs of the a2, a3, a4 parameters are indica­ ted in brackets

Fig. 9. Strehl numbers SN of the system after the /? angle-rotation o f the F element of the filter. For comparison, the Strehl number for the optimal polarizing filter reali­ zing a given Rayleigh resolution is presented

angle-rotation of the F element. F or comparison the Strehl number for the optimal polarizing filter realizing a given Eayleigh resolution is presented (in Fig. 3). Then the maximum of the encircled energy factor (for q = 1) and a given Eayleigh resolution of the system m ay be assured b y one polarizing filter. Only the adequate choice of the signs o f parameters a2, a3, a4 and the possibility o f the F element rotation are required.

Losses of the Strehl number are not high in relation to the optimal filter assuring a given Eayleigh resolution and m ay be additionally diminished when the maximum of the encircled energy factor is not demanded.

504 W . URBAftCZYK

Let us com pose the function

F C(d, ÔB) = K E e(à)q1+ 8 N B(a, dB)q2+ 8 N E(d)qa, where N 2 0JJcos(2ajr)e o B (2 a J ) K E e{d) = ^ =1 _N (

10

) (11) £ P J c o s 2( 2 aJ) 7 = 1

is the encircled energy factor (/3 = 0 ) ,

N 8 N B(d, dR ) = [ ] ? R K 1( 0 ) c o s ( 2 a J) c o s ( 2 /S(<3ii:)) / — 1 JV £ R % i( 0 ) s in ( 2 aj)sin(2/S(<5fl))j2

_{1 1 2}

)

7=1

is the Strehl number of the system after the 0 angle-rotation of the F element o f the filter, N 7=1

100

95 90 85 8^ e(0·) —

[ 2 PJ\rj(b)cos(2ai) |2

KEj.i[%] (13)

c

0°

30°-20

°

10

° --

10

° - -20°- -30°--to0· ... B2 r ' ---a2 ArBrL-Cj-0 6o 3.2 2.8 2A 2.0

M

...'VA3 —--- >B3 ' ... ^ VBi.

Fig. 10. Strehl number 8N E and the enircled energy factor K E e (for q = = 1) at tlie initial position of the filter (/? = 0) as the function of the Rayleigh resolution 8B of the system. A - = _1, q2 = 0, qs = 0; B

- 2i = t e p . q t = 1/2/2, q3 = 0; G — 91 = 0.98, g2 = 0, 23 = 0.07

Fig. 11. Dependence of the angles a1, a2, a3, a4 upon the Rayleigh resolution of the system SB . Symbol A i indicates parameter a,· in the

On polarizing filters application... 505

is the Strehl number of the system fo r the initial position of the F element {/J = 0 ) ; qx, q3, q3 are the weight factors useful to control the maximization of the E E e, SNB, 8 N E functions.

Carrying out the numerical maximization of the FG(a, dR) function, we will obtain the parameters a2 . .. aN, /? of polarizing filter. Properties of this filter, at the initial (/5 = 0) and final (/? =£ 0) positions depend on the factors qlf q3, q3. Exem plary calculations were made fo r N = 4 and q = 1. Eesults are shown in

Eigs. 10-13. -50° snr -70° _\\ 0.7 _{\ /3*0°} -90a \ a\ \ \ M \ \ \ _\ \ \ _{\ v} 0.5

: s \

\ \\\_{\ ' v '} \ ' \ X -110“ \ \ ' ' ''V ''s 'V B 0.3--130° ''C 6r 0.1 \ '»V n _V.-B --A C 6R 3A 1 30 ' 20 ' 2.2 3A ' 3JD ' 2.6 ‘ 2.2 Pig. 12. Rotation angle (l vs. the

Rayleigh resolution of the system dB

Pig. 13. Strehl number SN B vs. the Rayleigh resolution of the system

6R at the final position (/5 ^ 0) of the F element of the polarizing filter. For comparison the Strehl number, for the optimal polarizing filter realizing a given Rayleigh resolution, is presented

References

[1] Ba r a k a t R ., J. Opt. Soc. Am. 52 (1962), 264. [2] Ba r a k a t R., J. Opt. Soc. Am. 52 (1962), 276.

[3] Wil k in s J. E., J. Opt. Soc. Am. 53 (1963), 420.

[4] Wil k in s J. E., J. Opt. Soc. Am. 67 (1977), 1027.

[5] Sl e p ia n D., J. Opt. Soc. Am. 55 (1965), 1110.

Received June 30, 1983